Since \(H\)  is an \((n-k+\lceil \frac{n}{r}\rceil )\times (n+\lceil \frac{n}{r}\rceil )\)  matrix and the rows of \(H\)  are linearly independent from linear algebra, the dimension of \(\mathcal {C}\)  is \((n+\lceil \frac{n}{r}\rceil )-(n-k+\lceil \frac{n}{r}\rceil )=k\) .
From Lemma REF , any \(d-1\)  columns of \(H\)  are linearly independent.
Hence, the minimum distance of \(\mathcal {C}\)  is at least \(d\)  from [1]}.
Moreover, the locality of \(\mathcal {C}\)  is \(r\)  from the parity-check matrix \(H\)  and Corollary REF .

